Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems

Jose Anderson Cardoso; Patricio Cerda; Denilson Pereira; Pedro Ubilla

doi:10.3934/dcds.2020392

Article Contents

2021, Volume 41, Issue 6: 2947-2969. Doi: 10.3934/dcds.2020392

This issue Previous Article An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth Next Article On global well-posedness of the modified KdV equation in modulation spaces

Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems

1.
Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão-SE, 49100-000, Brazil
2.
Departamento de Matematica y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
3.
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande 58429-900, Brazil

^* Corresponding author
^* Corresponding author

Received: December 2019

Revised: October 2020

Early access: December 2020

Published: June 2021

The first author is partially supported by FAPITEC/CAPES and by CNPq - Universal.
The second author was partially supported by Proyecto código 042033CL, Dirección de Investigación, Científica y Tecnológica, DICYT.
The third author was partially supported by Proyecto código 041933UL POSTDOC, Dirección de Investigación, Científica y Tecnológica, DICYT.
The fourth author was partially supported by FONDECYT grant 1181125, 1161635, 1171691.

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Abstract

We prove the existence of a bounded positive solution for the following stationary Schrödinger equation

$ \begin{equation*} -\Delta u+V(x)u = f(x,u),\,\,\, x\in\mathbb{R}^n,\,\, n\geq 3, \end{equation*} $

where $ V $ is a vanishing potential and $ f $ has a sublinear growth at the origin (for example if $ f(x,u) $ is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [6]. In addition, if $ f $ has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type $ \rho(x)f(u) $ where $ f $ is a concave-convex function and $ \rho $ satisfies the $ \mathrm{(H)} $ property introduced in [6]. We also note that we do not impose any integrability assumptions on the function $ \rho $, which is imposed in most works.

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Mathematics Subject Classification: 35J20, 35J10, 35J91, 35J15, 35B09.

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References

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